Category

Published on

01 Jun 2010

Abstract

The Ensemble Monte Carlo technique has been used now for over 30 years as a numerical method to simulate nonequilibrium transport in semiconductor materials and devices, and has been the subject of numerous books and reviews. In application to transport problems, a random walk is generated to simulate the stochastic motion of particles subject to collision processes in some medium. This process of random walk generation may be used to evaluate integral equations and is connected to the general random sampling technique used in the evaluation of multi-dimensional integrals.

The basic technique is to simulate the free particle motion (referred to as the free flight) terminated by instantaneous random scattering events. The Monte Carlo algorithm consists of generating random free flight times for each particle, choosing the type of scattering occurring at the end of the free flight, changing the final energy and momentum of the particle after scattering, and then repeating the procedure for the next free flight.

Sampling the particle motion at various times throughout the simulation allows for the statistical estimation of physically interesting quantities such as the single particle distribution function, the average drift velocity in the presence of an applied electric field, the average energy of the particles, etc. By simulating an ensemble of particles, representative of the physical system of interest, the non-stationary time-dependent evolution of the electron and hole distributions under the influence of a time-dependent driving force may be simulated.

The particle-based picture, in which the particle motion is decomposed into free flights terminated by instantaneous collisions, is basically the same picture underlying the derivation of the semi-classical BTE. In fact, it may be shown that the one-particle distribution function obtained from the random walk Monte Carlo technique satisfies the BTE for a homogeneous system in the long-time limit.